有限域上Carlitz方程的解数

The Number of Solutions of Carlitz's Equation

设F_q是含有q个元素的有限域,其中q=p^t,t≥1,p是一个奇素数.研究了Carlitz方程的推广形式(a_1x_1^(m_1)+…+a_nx_n^(m_n)+a_(n+1)x_(n+1)^(m_(n+1))+…+a_(n+s)x_(n+s)^(m_(n+s)))~k=bx_1^(k_1)…x_n^(k_n),其中ai,b∈F_q~*,s≥1,n≥1.当方程变量的指数满足一定条件时,得到了方程的解数公式.

Let Fq be a finite field of q elements,whereq=pt,t≥1,pis an odd prime number.We consider the generalized Carlitz’s equation a1xm11+…+anxmnn+an+1xmn+1n+1+…+an+sxmn+sn+sk=bxk11…xknn,where a,b∈F*q,s≥1,n≥1.When the exponents of the equation satisfy certain conditions,we obtain the formula for the number of solutions.